Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote the $\infty$-category of small stable $\infty$-categories and exact functors by $Cat^{\rm ex}_{\infty}$. Suppose I have given a functor $$ F \colon \mathcal{I} \times \mathcal{I}^{\rm op} \to Cat^{\rm ex}_{\infty} $$ where I am secretly regarding the category on the left as an $\infty$-category.

Are coends defined for $\infty$-functors as above? Does the coend of $F$ as above always exist?

If the answer to the above questions is "yes": Can we describe the weak Kan complex underlying the coend in some way?